Function Composition Exploration

Function Composition Exploration

Read each question carefully and provide clear, complete answers. Show all your work where applicable.

1. The notation (f ∘ g)(x) represents the   of functions f and g.

2. To evaluate (f ∘ g)(x), you first apply the function   to x, and then apply the function   to the result.

3. The domain of (f ∘ g)(x) consists of all x in the domain of g such that g(x) is in the domain of  .

Given f(x) = 2x + 3 and g(x) = x² - 1, find the following:

4. (f ∘ g)(2)

5. (g ∘ f)(-1)

6. (f ∘ f)(0)

Given f(x) = 3x - 2 and g(x) = x + 5, which of the following is (f ∘ g)(x)?

Given h(x) = x² and k(x) = √x, which of the following is (k ∘ h)(x)?

Given f(x) = 1/(x-2) and g(x) = √x, find the composite function and its domain.

7. Find (f ∘ g)(x).

8. State the domain of (f ∘ g)(x).

For any two functions f and g, (f ∘ g)(x) is always equal to (g ∘ f)(x).

If f(x) = x and g(x) = x², then (f ∘ g)(x) = x².

A store offers a 20% discount on all items. On top of that, a coupon gives an additional $10 off. Let x be the original price of an item.

9. Write a function D(x) for the price after the 20% discount.

10. Write a function C(x) for the price after the $10 coupon.

11. Find (C ∘ D)(x) and explain what it represents in this context.

12. Find (D ∘ C)(x) and explain what it represents in this context.

13. Which order of applying the discounts results in a lower final price for an item that originally costs $100? Justify your answer.